The approximation of the metric terms \(J^{v}\frac{\partial \xi _{j}}{\partial x_{k}}\)  at the
combined volume and surface quadrature points is denoted with the
diagonal matrix \(\mathbf {G}_{jk}\) .
As noted above, these are often approximated using the discrete operators. One
benefit of this is that the volume \(\mathbf {G}_{jk}^{v}\)  and surface
\(\mathbf {G}_{jk}^{f}\)  can be defined in a consistent manner, e.g.,
\(\mathbf {G}_{jk}^{f} = \operatorname{Diagonal}\left(\mathbf {E}^{v}\operatorname{diag}\left(\mathbf {G}_{jk}^{v}\right)\right)\)  where \(\operatorname{diag}\)  extracts the diagonal
of a matrix as a vector.
In the discrete entropy stability analysis we will require that the metrics
discretely satisfy the geometric conservation law [1]}, [2]}
\(\sum _{k=1}^{d} \mathbf {Q}_{k} \mathbf {G}_{jk} \mathbf {1} = \mathbf {0},\) 
